The homotopy theory of small categories was mainly founded by Quillen in his quest for a definition of higher algebraic K-theory. Quillen in particular showed his famous theorems A and B, and the fact that the homotopy category of small categories (endowed with the weak equivalences induced by the nerve functor) is equivalent to the homotopy category of spaces. This equivalence of homotopy categories was then promoted to a Quillen equivalence by Thomason.

In this talk based on joint work with Georges Maltsiniotis, I will present some ideas on the homotopy theory of strict n-categories. In particular, I will explain a conditional proof of the existence of a Thomason model structure on strict n-categories. For n = 1 and n = 2, we will deduce this existence unconditionally.

If a Poisson structure on a manifold $M$ vanishes at a point $x$, then its linearization is a Poisson structure on the tangent space $T_x M$. The Poisson structure on $M$ is called linearizable at $x$ if there is a germ of a Poisson diffeomorphism between $M$ and $T_x M$.

Let $M$ be a model category and $F∶ M \to \underline{\mathrm{ModCat}}$ a relative functor. If $F$ satisﬁes a "strengthened relativeness" condition, we deﬁne a model structure on the Grothendieck construction $\int_M F$ called the integral model structure. This model structure is “homotopy-invariant” in that for two such functors $F, F': M \to \underline{\mathrm{ModCat}}$, a pseudo-natural transformation $\Phi ∶ F \Rightarrow F'$ which is an object-wise Quillen equivalence induces a Quillen equivalence between the corresponding integral structures:$$\int_M F \simeq \int_M F'$$

As applications, we describe the integral model structure that arises from several functors. These include the functor $B∶ sGp \to \underline{\mathrm{ModCat}}$ given by the projective model structure on $G$-spaces, $G \mapsto S^G$ and the functor $(\bullet)-\mathrm{LMod}: \mathrm{Alg}(C) \to \underline{\mathrm{ModCat}}$ given by the model structure of (left) modules over an algebra object in a suitable monoidal model category $C$, satisfying the assumptions of Schwede and Shipley. This is joint work with Y. Harpaz.

We explain that the BRST complex of closed string field theory is an algebra over the bar construction (called in this context the Feynman transform) of the modular envelope of the operad Com for commutative associative algebras. This algebraic structure is sometimes called a loop or quantum homotopy Lie algebra. We then discuss an analogous result for open strings. Here the central problem lies already in describing the modular envelope of the operad Ass for associative algebras. We show that this task has an interesting geometric content related to the cow's stomach.

There is a canonical way to assign to each loop on a Riemannian manifold a Hilbert space with a conformal net that acts on it. The failure of these Hilbert spaces to constitute a locally trivial bundle over loop space is measured by the curvature of the metric. We use higher geometry (2-groups and 2-bundles) to describe this situation and outline why the Connes fusion product of Hilbert spaces should extend to a fusion structure for the spinor bundle on loop space. This is work in progress.

I will talk about a special class of submanifolds which play a role in Poisson geometry akin to that of transversals in foliation theory and to that of symplectic submanifolds in Symplectic geometry. The discussion will revolve around normal forms, stability and (non-)existence results.

For many applications it is important to know if there exists a transferred model structures for ``monoid-like'' objects in monoidal model categories. These include genuine monoids, but also all kinds of operads, e.g. symmetric, cyclic, modular, n-operads, dioperads, properads and (wheeled) PROP's etc. All these structures can be realised as algebras over polynomial monads.

In my talk I will explain a general condition for a polynomial monad called tameness which ensures the existence and left properness of a transferred model structure for its algebras in an h-monoidal model category. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame. Many important polynomial monads are shown to be tame. On the other hand there are interesting polynomial monads which are not tame. For example, monads for modular operads or PROPs. We show that failure of tameness condition can be used to find obstructions for the existence of transferred model structure on algebras.